All the principles we have explained above were intended so that you will be ready and prepared to know [how] to sight [the moon]. When you desire to know [how to sight the moon on a particular night], calculate the true position of the sun, the true position of the moon, and the position of the head [of the moon's orbit] for the time of the sighting [of the moon].

Afterwards, subtract the position of the sun from the position of the moon. The remainder is referred to as the first longitude.

After having determined the position of the head and the moon's position,1 you will be able to determine the moon's latitude and whether this latitude is northerly or southerly. This figure is referred to as the first latitude. Be careful with regard to [these two symbols,] the first longitude and the first latitude, and have them at hand [for later calculations].

Consider the first longitude:2 If the figure you arrive at is equal to nine degrees or less, know that it will definitely be impossible for the moon to be sighted on that night throughout Eretz Yisrael; no other calculation is necessary.

If the first longitude is more than fifteen degrees, know that the moon will definitely be sighted throughout Eretz Yisrael; no other calculation is necessary.3

If the first longitude is between nine and fifteen degrees, it will be necessary for you to make further calculations to know whether the moon will be sighted or not.

When does the above apply? When the true position of the moon is located [in the area] between the beginning of the constellation of Capricorn and the end of the constellation of Gemini.4 If, however, the position of the moon [is located in the range] between the beginning of the constellation of Cancer and the end of the constellation of Sagittarius, and the first longitude is ten degrees or less,5 know that it will definitely be impossible for the moon to be sighted on that night throughout Eretz Yisrael; no other calculation is necessary.

If the first longitude6 is more than twenty-four degrees, know that the moon will definitely be sighted throughout Eretz Yisrael; no other calculation is necessary.

If the first longitude is between ten and twenty-four degrees, it will be necessary to make further calculations to know whether or not the moon will be sighted.7

These are the [further] calculations [necessary to determine] the sighting of the moon: Consider the constellation in which the moon is located.8 If it is the constellation of Aries, subtract9 59 minutes from the first longitude. If it is the constellation of Taurus, subtract one degree from the longitude.10 If it is the constellation of Gemini, subtract 58 minutes from the longitude.11 If it is the constellation of Cancer, subtract 5212 minutes from the longitude.

If it is the constellation of Leo, subtract 43 minutes from the longitude.13 If it is the constellation of Virgo, subtract 37 minutes from the longitude. If it is the constellation of Libra, subtract 34 minutes from the longitude.

If it is the constellation of Scorpio, subtract 34 minutes from the longitude. If it is the constellation of Sagittarius, subtract 36 minutes from the longitude.14 If it is the constellation of Capricorn, subtract 44 minutes from the longitude.15 If it is the constellation of Aquarius, subtract 53 minutes from the longitude. If it is the constellation of Pisces, subtract 58 minutes from the longitude.16The remainder after these minutes have been subtracted from the longitude is referred to as the second longitude.

Why are these minutes subtracted? Because the true position of the moon is not the place where the moon will actually be sighted [in the sky]. Instead, there is a [small] difference in both longitude and latitude.17 This [difference] is referred to as the sighting adjustment.

The sighting adjustment for the moon's longitude at the hour of the sighting of the moon18 should always be subtracted from the longitude, as we explained.

[The latter point is not necessarily true,] by contrast, with regard to the sighting adjustment for the moon's latitude.19If the moon's latitude is northerly, we subtract the minutes of the sighting adjustment for the moon's latitude from the moon's first latitude.20 If, however, the moon's latitude is southerly, we add the minutes of the sighting adjustment for the moon's latitude to the moon's first latitude.21 The result after the addition or subtraction of these minutes to or from the [moon's] first latitude is referred to as the second latitude.

How many minutes are added or subtracted? If the moon is in the constellation of Aries, 9 minutes.22 If it is in the constellation of Taurus, 10 minutes. If it is in the constellation of Gemini, 16 minutes. If it is in the constellation of Cancer, 27 minutes. If it is in the constellation of Leo, 38 minutes. If it is in the constellation of Virgo, 44 minutes.

If it is in the constellation of Libra, 46 minutes.23 If it is in the constellation of Scorpio, 45 minutes. If it is in the constellation of Sagittarius, 44 minutes. If it is in the constellation of Capricorn, 36 minutes. If it is in the constellation of Aquarius, 2724 minutes. If it is in the constellation of Pisces, 12 minutes.

Since you know the [number of] minutes [for the adjustment] for each constellation, add or subtract them to or from the first latitude, as was explained, and arrive at the second latitude. You already know whether it will be northerly or southerly,25 and you will know the number of degrees and minutes of this second latitude. Prepare this figure and have it at hand.

Afterwards,26 you must set aside [for later use in subtracting or adding] a portion of this second latitude, because the moon fluctuates slightly in its orbit.27 What is the size of the portion you must separate? [This depends on the position of the moon in the celestial sphere.]28

If the moon is located between the beginning of the constellation of Aries and its twentieth degree - and similarly, [when the moon is located] between the beginning of the constellation of Libra and its twentieth degree - separate two fifths of the second latitude.29

If the moon is located between the twentieth degree of the constellation of Aries and the tenth degree of the constellation of Taurus - and similarly, [when the moon is located] between the twentieth degree of the constellation of Libra and the tenth degree of the constellation of Scorpio - separate one third from the second latitude.

If the moon is located between the tenth degree of the constellation of Taurus and its twentieth degree - and similarly, [when the moon is located] between the tenth degree of the constellation of Scorpio and its twentieth degree - separate one fourth from the second latitude.

If the moon is located between the twentieth degree of the constellation of Taurus and its end - and similarly, [if the moon is located] between the twentieth degree of the constellation of Scorpio and its end - separate one fifth from the second latitude.

If the moon is located between the beginning of the constellation of Gemini and its tenth degree - and similarly, [if the moon is located] between the beginning of the constellation of Sagittarius and its tenth degree - separate one sixth of the second latitude.

If the moon is located between the tenth degree of the constellation of Gemini and its twentieth degree - and similarly, [if the moon is located] between the tenth degree of the constellation of Sagittarius and its twentieth degree - separate one twelfth from the second latitude.

If the moon is located between the twentieth degree of the constellation of Gemini and its twenty-fifth degree - and similarly, [if the moon is located] between the twentieth degree of the constellation of Sagittarius and its twenty-fifth degree - separate one twenty-fourth from the second latitude.

If the moon is located between the twenty-fifth degree of the constellation of Gemini and the fifth degree of the constellation of Cancer - and similarly, [if the moon is located] between the twenty-fifth degree of the constellation of Sagittarius and the fifth degree of the constellation of Capricorn - do not make any separation, because at this point the moon does not fluctuate at all from its orbit.30

If the moon is located between the fifth degree of the constellation of Cancer and its tenth degree - and similarly, [if the moon is located] between the fifth degree of the constellation of Capricorn and its tenth degree - separate one twenty-fourth from the second latitude.

If the moon is located between the tenth degree of the constellation of Cancer and its twentieth degree - and similarly, [if the moon is located] between the tenth degree of the constellation of Capricorn and its twentieth degree - separate one twelfth from the second latitude.

If the moon is located between the twentieth degree of the constellation of Cancer and its end - and similarly, [if the moon is located] between the twentieth degree of the constellation of Capricorn and its end - separate one sixth from the second latitude.

If the moon is located between the beginning of the constellation of Leo and its tenth degree - and similarly, [if the moon is located] between the beginning of the constellation of the Aquarius and its tenth degree - separate one fifth of the second latitude.

If the moon is located between the tenth degree of the constellation of Leo and its twentieth degree - and similarly, [if the moon is located] between the tenth degree of the constellation of Aquarius and its twentieth degree - separate one fourth from the second latitude.

If the moon is located between the twentieth degree of the constellation of Leo and the tenth degree of the constellation of Virgo, and similarly, [if the moon is located] between the twentieth degree of the constellation of Aquarius and tenth degree of the constellation of Pisces - separate one third from the second latitude.

If the moon is located between the tenth degree of the constellation of Virgo and its end - and similarly, [if the moon is located] between the tenth degree of the constellation of Pisces and its end - separate two fifths from the second latitude.

This portion that is separated from the second latitude is referred to as the circuit of the moon.

Afterwards, go back and consider whether the latitude of the moon is northerly or southerly. If it is northerly, subtract [the adjustment referred to as] the circuit of the moon from the second longitude.31 If the moon's longitude is southerly, add the circuit of the moon to the second longitude.32

When does the above apply? When the moon's position is located between the beginning of the constellation of Capricorn and the end of the constellation of Gemini.33 If, however, the moon's position is located between the beginning of the constellation of Cancer and the end of the constellation of Sagittarius,34 the opposite is true: If the moon's latitude is northerly, the circuit should be added to the second longitude,35 and if the moon's latitude is southerly, the circuit should be subtracted from the second longitude.36

The remainder after the additions or subtractions have been made to the second longitude is referred to as the third longitude. Know that if there is no fluctuation within the circuit, and there is no figure to be separated from the second latitude,37 the second longitude also will serve as the third longitude, without any decrease or increase.

Afterwards, go back and see the constellation in which the third longitude - i.e., [the amended figure representing] the distance between the sun and the moon - is located:38 If it is located in the constellation of Pisces or Aries, add one sixth [of its length] to the third longitude.39 If the [third] longitude is located in the constellation of Aquarius or the constellation of Taurus, add one fifth [of its length] to the third longitude.40 If the [third] longitude is located in the constellation of Capricorn or the constellation of Gemini, add one sixth [of its length] to the third longitude.

If the [third] longitude is located in the constellation of Sagittarius or the constellation of Cancer, leave the third longitude as it is, without making any addition or subtraction.41 If the [third] longitude is located in the constellation of Scorpio or the constellation of Leo, subtract one fifth [of its length] from the third longitude. If the [third] longitude is located in the constellation of Libra or the constellation of Virgo, subtract one third [of its length] from the third longitude.42

The figure resulting from these subtractions or additions to the third longitude, or from leaving it without adjustment, is referred to as the fourth longitude.

[Afterwards, a further correction is necessary:] Return to the first latitude,43 and set aside two thirds [of its length]. This is called the correction [resulting from geographic] latitude.44

Consider whether the latitude of the moon is northerly. If so, add the correction [resulting from geographic] latitude to the fourth longitude.45 If the latitude of the moon is southerly, subtract the correction [resulting from geographic] latitude from the fourth longitude.46 The figure resulting from these subtractions or additions to the fourth longitude is referred to as the arc of sighting.

What is implied? For example, let us attempt to determine whether or not it will be possible to sight the moon on Friday night, the second of Iyar of this year: First, it is necessary to determine the true position of the sun, the true position of the moon, and the moon's latitude for this time47 according to the methods that were disclosed [in the previous chapters].

The result is that the true position of the sun is seven degrees and nine minutes in the constellation of Taurus, in symbols 7° 9'. The true position of the moon is eighteen degrees and thirty-six minutes in the constellation of Taurus, in symbols 18° 36'. The moon's latitude is three degrees and fifty-three minutes, in symbols 3° 53', and it is southerly. This is the first latitude.

Afterwards, you should subtract the position of the sun from the position of the moon, arriving at a remainder of eleven degrees and twenty-seven minutes, in symbols 11° 27'. This is the first longitude.

Since the moon is located in the constellation of Taurus, the sighting adjustment for the longitude will be one degree. This should be subtracted from the first longitude, producing a second longitude of 10 degrees and twenty-seven minutes, in symbols 10° 27'. The sighting adjustment for the latitude is ten minutes. Since the moon's latitude is southerly, this sighting adjustment of ten minutes should be added to the moon's latitude, producing a second latitude of four degrees and three minutes, in symbols 4° 3'.

Since the moon is located in the eighteenth minute of the constellation of Taurus, it is proper to set aside one fourth of the second latitude as the circuit of the moon. Thus, at this time, the circuit of the moon will be one degree and one minute. We pay no attention to the seconds.

Since the latitude of the moon is southerly and the true position of the moon is located between the beginning [of the constellation] of Capricorn and the beginning [of the constellation] of Cancer, it is correct to add the circuit of the moon to the second longitude. Thus, the third longitude will be eleven degrees and twenty-eight minutes, in symbols 11° 28'.

Since this longitude is located in the constellation of Taurus, it is fitting to add one fifth to the third longitude - i.e., two degrees and eighteen minutes. Thus, the fourth longitude will be thirteen degrees and forty-six minutes, in symbols 13° 46'.

We then go back to the first latitude and separate two thirds of it. Thus, the correction [resulting from geographic] latitude is two degrees and thirty-five minutes. Since the latitude is southerly, the correction [resulting from geographic] latitude should be subtracted from the fourth longitude, leaving a remainder of eleven degrees and eleven minutes, in symbols 11° 11'. This is the arc of sighting on this night.

Following this process, you can determine the number of degrees and the number of minutes every night for the moon's sighting48, which you desire for all time.

After you have determined this arc [of sighting], consider it. Know [these rules]: If the arc of sighting is nine degrees or less, it is impossible for the moon to be sighted anywhere in Eretz Yisrael. If the arc of sighting is more than fourteen degrees, it is impossible49 for it not to be seen and openly revealed throughout Eretz Yisrael.

If the arc of sighting is between the beginning of the tenth degree and the end of the fourteenth degree, [the following procedure should be followed:] One should consider the arc of sighting in relation to the first longitude50 and determine whether or not the moon will be seen from the limits prevalent [at that time]. They are referred to as the sighting limits.

The following are the sighting limits: If the arc of sighting is between nine and ten degrees, or more than ten [degrees], and the first longitude is thirteen degrees or more, [the moon] will surely be sighted. If the arc is less than this, or if the first longitude is less than this, it will not be sighted.

If the arc of sighting is between ten and eleven degrees, or more than eleven [degrees], and the first longitude is twelve degrees or more, [the moon] will surely be sighted. If the arc is less than this, or if the first longitude is less than this, it will not be sighted.

If the arc of sighting is between eleven and twelve degrees, or more than twelve [degrees], and the first longitude is eleven degrees or more, [the moon] will surely be sighted. If the arc is less than this, or if the first longitude is less than this, it will not be sighted.

If the arc of sighting is between twelve and thirteen degrees, or more than thirteen [degrees], and the first longitude is ten degrees or more, [the moon] will surely be sighted. If the arc is less than this, or if the first longitude is less than this, it will not be sighted.

If the arc of sighting is between thirteen and fourteen degrees, or more than fourteen [degrees], and the first longitude is nine degrees or more, [the moon] will surely be sighted. If the arc is less than this, or if the first longitude is less than this, it will not be sighted. This concludes the sighting limits.

What is implied? If we were to consider the arc of sighting for Friday night, the second of Iyar of this year, we would calculate the arc of sighting to be eleven degrees and eleven minutes, as you have already determined.51

Since the arc of sighting is between ten and fourteen degrees, it is necessary to consider it in relation to the first longitude. It has already been established that on this night, the [first] longitude is ten degrees and twenty seven minutes. Since the first longitude is greater than eleven [degrees], we can be assured that the moon will be sighted on that night according to the limits established. Similar procedures should be followed [on every occasion when it is necessary to consider] the arc of sighting in relation to its first longitude.

From all the above, you have seen the extent of the calculations, and the additions and the subtractions that required much effort to present a method that comes close [to being exact]52 without necessitating extremely complicated calculations. [This process is necessary] because the moon has major incongruities in its orbit.

In this vein, our Sages said,53 "'The sun knows the time of its setting'; the moon does not know the time of its setting." Similarly, our Sages said,54 "At times, its setting is prolonged, and at times, it is hastened." This is reflected in these calculations, where at times it is necessary to add, and at times it is necessary to subtract until we arrive at the arc of sighting. And as explained, at times the arc of sighting is great, and at times it is small.

The rationales for all these calculations, and the reasons why this number is added, and why that subtraction is made, and how all these concepts are known, and the proofs for each of these principles are [the subject] of the wisdom of astronomy and geometry, concerning which the Greeks wrote many books.

These texts are presently in the hands of the sages. The texts written by the Sages of Israel in the age of the prophets from the tribe of Yissachar55 have not been transmitted to us. Nevertheless, since these concepts can be proven in an unshakable manner, leaving no room for question, the identity of the author, be he a prophet or a gentile, is of no concern.56 For a matter whose rationale has been revealed and has proven truthful in an unshakable manner, we do not rely on [the personal authority of] the individual who made these statements or taught these concepts, but on the proofs he presented and the reasons he made known.57

In this instance as well, our translation is based on authentic manuscript editions of the Mishneh Torah and early printings. There is a printing error in the standard published text, where the words "and the first latitude" were added unnecessarily.

In these months, the ecliptic (the plane of the sun's orbit as extended to the celestial sphere) is inclined to the north. After the conjunction, the moon proceeds away from the position of the sun. When the inclination of the ecliptic is northward, this movement places it in a more northerly position. Therefore, the moon will set later than would be foreseen otherwise, resulting in a greater possibility of seeing the new moon.

As mentioned in the notes of Chapter 16, as the longitude of the moon increases, seeing the moon also becomes easier. In these months, however, a lesser longitude is required.

In these months, the ecliptic is inclined to the south. As the moon proceeds away from the position of the sun after conjunction, it will be in a more southerly position in these months. Therefore,the moon will set earlier than would be foreseen otherwise, resulting in a lesser possibility of seeing the new moon. To compensate for this difference, a greater longitude is required.

Our translation is based on authentic manuscript editions of the Mishneh Torah and early printings. There is a printing error in the standard published text, and the word "latitude" was added unnecessarily.

To summarize the Rambam's statements to this point: When the longitude of the moon (the angular distance between the moon and the sun) is minimal, the moon's crescent will be small and the interval between the time of its setting and that of the sun will be small. Hence, it is unlikely that the moon will be sighted.

When the longitude of the moon is greater, the size of the moon's crescent will increase, as will the interval between the time of its setting and that of the sun. Accordingly, the possibility of sighting the moon will increase.

When the longitude is significantly large, it is obvious that the moon will be seen and no other calculations are necessary. When, however, the longitude is of intermediate size, there is a question whether the moon will be seen. The resolution of this question depends on the inclination of the ecliptic and the latitude of the moon - i.e., the angle - and the direction of that angle - to which the moon is inclined from the plane of the sun.

[In this context, it is worthy to mention a question raised by several contemporary commentaries on Hilchot Kiddush HaChodesh: Seemingly, there is a direct connection between the concepts mentioned at the conclusion of Chapter 15 and those mentioned at the beginning of this chapter. Why does the Rambam interpose the discussion of the moon's latitude (the subject matter of Chapter 16, which becomes relevant only in subsequent halachot) between them?]

The sighting adjustment for longitude is based on two different factors: a) whether the constellation is inclined to the north or to the south as it intersects the horizon of Jerusalem, and b) the extent of the southerly position of that constellation.

To explain: The constellations intersect the horizon at different angles, reflecting the pattern of their inclination in the heavenly sphere. The constellations from Capricorn until Gemini intersect the horizon at a northerly angle, and the constellations from Cancer to Sagittarius intersect the horizon at a southerly angle.

With regard to the second factor, all the constellations are located to the south of Jerusalem. Jerusalem is located 32 degrees north, and the constellation of Cancer, the most northerly of the constellations, is located 23 1/2 degrees north. The more northerly a constellation is located, however, the greater the need for a subtraction from its longitude.

This is the constellation that the moon enters at the vernal (spring) equinox. It is inclined to the north and is not located in an extremely southerly position. Hence, a large sighting adjustment is necessary.

The commentaries have noted that although the general thrust of the adjustments suggested by the Rambam conform to the calculations of the astronomers, the exact figures he gives follow neither the classic Greek figures nor those of modern astronomy. It is possible to explain that the Rambam was speaking merely in approximations, giving us a figure useful enough to calculate the position where the moon would be sighted, but not an exact scientific measure. This theory is borne out by the fact that he does not provide different measures for northern and southern latitudes, although according to science these figures vary.

To explain: The true position of the moon reflects the line extending from the center of the earth through the center of the moon, as it is projected against the heavenly sphere. Since Jerusalem (or for that matter, any other location on the earth's surface) is not located at the center of the earth, but rather 4000 miles away, there will be a slight difference between the line described previously and the line extending from a person standing in Jerusalem to the center of the moon, as it is projected against the heavenly sphere. The closer the moon is to the horizon, the larger the sighting adjustment that has to be made.

[The same concept applies with regard to the sun. Nevertheless, since the distance between the earth and the sun is great, the angular difference between these two lines is not of consequence. The moon, by contrast, is located much closer to the earth and, at times, a difference of close to a degree can arise.]

In the evening, the moon will always appear slightly closer to the horizon than it actually is - i.e., it will appear closer to the position of the sun. Therefore, the angular difference between the two lines mentioned above should be subtracted from the moon's true position. As explained above, the extent of the adjustment to be made depends on the inclination at which constellation intersects the horizon and its latitude in the heavenly sphere.

The sighting adjustment for the moon's latitude is derived by creating a parallax - i.e., a line directly parallel to the line running from the point of the moon's first longitude to its first latitude is drawn from the point of its second longitude. A second line is drawn from the position of an onlooker in Jerusalem through the point of the first latitude and intersecting the line of the moon's second latitude. The point where these two lines intersect is the moon's second longitude. The adjustment mentioned in the following halachah represents the angle between these two lines.

Because Jerusalem is situated in a more northerly position than all the Zodiac constellations, the moon will always appear more southerly than it actually is. Therefore, if its latitude is northerly, a subtraction is necessary. To use geometric terms: When the moon's latitude is northerly, its second latitude will always be closer to the point of its longitude than to its first latitude.

Since the moon will always appear more southerly, an addition is required when its original latitude is southerly. In geometric terms: When the moon's latitude is southerly, its second latitude will always be further removed from the point of its longitude than its first latitude.

The purpose of the calculations that follow (reaching a third longitude and a fourth longitude) is to calculate the time between the setting of the sun and the setting of the moon. The first longitude is sufficient to inform us whether or not the crescent of the moon will be large enough to be visible. The subsequent calculations are necessary to determine whether or not there will be sufficient time for actually sighting the moon. For when the crescent is small, it is difficult to detect unless there is ample time before it sets.

The third longitude reflects the point in the celestial sphere that will set at the same time as the moon does, as seen by a person standing on the equator. This is not the point in the celestial sphere where the moon appears to be located, but rather a point in the celestial sphere that is reached by drawing a line originating at the equator, running parallel to the horizon of the equator, and extending through the center of the moon. The point where this line intersects the celestial sphere is the third longitude.

The reason for associating the moon's position with the equator is to establish a connection with a standard measure of time. In Jerusalem (and for that matter, anywhere else in the northern or southern hemisphere), the apparent movement of the celestial sphere varies with the seasons. On the equator, by contrast, the movement of the celestial sphere is constant at all times, 15 minutes to the hour.

The Rambam's intent in these sets of calculations is to reach a point on the equator that will set at the same time the third longitude sets in Jerusalem. For although the third longitude was able to relate the moon's position to the equator, it did not take into consideration the difference between the horizon of the equator and the horizon of Jerusalem. This is accomplished by drawing a line from the third longitude to the equator, which is parallel to the horizon of Jerusalem.

Two factors are significant in determining the fourth longitude: a) The angle of the constellation's inclination to the horizon of the equator. The greater the inclination of the constellation, the closer the fourth longitude will be located to the equator.

b) whether the constellation is inclined to the north or to the south.

If the constellation is inclined to the north, the third longitude, and hence the place on the equator parallel to it, will be located further away from the horizon, resulting in a later setting and thus an extended fourth longitude. Conversely, if the inclination is southerly, the third longitude will be located closer to the horizon, resulting in a shortened fourth longitude.

Of these two factors, the latter is more significant, and causes a larger correction. To explain these factors with regard to the constellations of Pisces and Aries: These constellations are inclined to a great degree, a factor that would reduce the fourth longitude. Since, however, they are northerly inclined, and this is the stronger factor, a modest increase is required.

In this instance, the degree of inclination of these constellations is great and their inclination is southerly. Both of these factors lead to a reduction in the fourth longitude. Hence, the greatest subtraction is required.

The Ralbach questions why the Rambam refers to the first latitude. Seemingly, it would be appropriate to make this correction based on the second latitude, for there is a significant difference between it and the first latitude. According to trigonometry, it also would appear that the calculations should be based on the second latitude.

Although the fourth longitude established a relationship between the equator and Jerusalem, it is still dependent on the third longitude, which relates to the moon and the celestial sphere as they set on the horizon of the equator. Through the correction mentioned here, we find a place on the extension of the equator that will set at the same as the moon sets in Jerusalem. Having reached this point, we can calculate the difference in time (15 degrees to the hour) between the setting of the sun and this point (which will set at the same time as the setting of the moon). Accordingly, we will be able to determine whether or not this interval will allow for the sighting of the moon.

The correction for geographic longitude is reached by drawing a line from the position of the moon parallel to the horizon of Jerusalem. One might ask: If this was the Rambam's intent, why were so many intermediate steps - the definition of the second, third, and fourth latitudes - necessary? Why didn't he suggest drawing the above- mentioned line at the very beginning of his calculations?

The explanation is that the Rambam allowed an individual to follow his own steps in arriving at this final figure. I.e., these lines and distances are all artificial and can be determined only by calculations. Through trigonometry, if one knows the length of one side of a triangle and two angles, or the length of two sides and one angle, it is possible to calculate the size of all three angles and all three sides. To find the line extending from the moon to the equator parallel to the horizon of Jerusalem, the Rambam had to build sets of triangles, and calculate angles based on the relationship of one triangle to another. The process he followed is reflected in the series of corrections he offers.

A northerly latitude means that the actual position of the moon is further removed from the horizon than the third longitude. This will result in a later setting of the moon. Accordingly, the correction based on geographic latitude will require addition to the fourth longitude. This applies regardless of whether the inclination of the constellation in which the moon is located is northerly or southerly.

A southerly latitude means that the actual position of the moon is closer to the horizon than the third longitude. This will result in an earlier setting of the moon. Accordingly, the correction based on geographic latitude will require subtraction from the fourth longitude. This applies regardless of whether the inclination of the constellation in which the moon is located is northerly or southerly.

It is possible that the Rambam's wording alludes to a concept mentioned previously, that the calculations he suggests are applicable only at the beginning of the month, when the new moon might be sighted.

As mentioned at the beginning of this chapter, the first longitude gives us information regarding the size of the moon's crescent and the difference between the moon's setting and that of the sun. When the first longitude is sufficiently large or when it is sufficiently small, it is possible to determine whether or not the moon will be sighted without considering extenuating factors - e.g., its longitude, the inclination of the constellation in which it is located, and the extent of that inclination. When, however, the first longitude is of intermediate length, these extenuating factors must be considered. The establishment of a systematic method of considering these factors is the purpose of all the computations mentioned in this chapter.

As the Rambam mentioned at the very beginning of this discussion (Chapter 11, Halachah 6), the figures that he gives are not exact. They do, however, give us sufficient information to determine when and where the moon will be sighted.

Commenting on I Chronicles 12:32, "From the descendants of Yissachar, men who had understanding of the times...," Bereshit Rabbah 72:5 explains that the sages of the tribe of Yissachar were those responsible for the determination of the calendar. (See also the commentary of the Radak on this verse.)

The context of this commentary is not a proper place for a full discussion of the Rambam's perspective on the supposed conflicts between science and the Torah. It must be noted, however, that the statements made here, emphasizing the importance of the empirical evidence of science, should not be interpreted as indicating that the perspective science adopts at any given time should be accepted in place of the Torah's teachings. In this context, it is worthy to quote the Rambam's statements in Hilchot Shechitah 10:13:

Similarly, with regard to the conditions that we have enumerated as causing an animal to be trefah (unable to live for an extended period): Even though it appears from the medical knowledge available to us at present that some of these conditions are not fatal... all that is significant to us is what our Sages said, as [implied by Deuteronomy 17:11]: "[You shall act] according to the instructions that they will give you."

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